c

W=W

c

W>W

c

ergodic (multi)fractal Wigner-Dyson RM Power-law Banded RM P. W. Anderson, Phys. Rev.

109

, 1492 (1958) localized Banded RM 3

Outline

1. Power-law banded matrices and ultrametric ensemble: universality of the fractal dimensions 2. Fractal dimensions: beyond the universality 3. Criticality in 2D long-range hopping model 4. Dynamical scaling: validity of Chalker’s ansatz 4

Fractal dimensions

Moments: Extended states: Anomalous scaling exponents: Critical point: How one can calculate ? Green’s functions: Localized states: 5

I

q

= X

r

jÃ

n

(r )j

2q

® / L

¡ d q ( q¡ 1)

Power-law banded random matrices

Gaussian distributed, independent critical states at all values of mapping onto the non-linear σ-model weak multifractality almost diagonal matrix strong multifractality 6 A. D. Mirlin et. al., Phys. Rev. E

54

, 3221 (1996)

Ultrametric ensemble

Random hopping between boundary nodes of a tree of K generations with coordination number 2 Distance number of edges in the shortest path connecting i and j -- ultrametric Strong triangle inequality: Y. V. Fyodorov, AO, A. Rodriguez, J. Stat. Mech., L12001 (2009) 7

Almost diagonal matrices

If , then the moments can be calculated perturbatevely. determines the nataure of eigenstates in the thermodynamic limit localized states extended states critical states 8

Strong multifractality in the ultrametric ensemble

General expression: Ultrametric random matrices: Fractal dimensions: Y. V. Fyodorov, AO, A. Rodriguez, J. Stat. Mech., L12001 (2009) 9

Universality of fractal dimensions

Power-law banded matrices: Ultrametric random matrices:

universality

Y. V. Fyodorov, AO, A. Rodriguez, J. Stat. Mech., L12001 (2009) A. D. Mirlin and F. Evers, Phys. Rev. B

62

, 7920 (2000) 10

Outline

1. Power-law banded matrices and ultrametric ensemble: universality of the fractal dimensions 2. Fractal dimensions: beyond the universality 3. Criticality in 2D long-range hopping model 4. Dynamical scaling: validity of Chalker’s ansatz 11

Fractal dimensions: beyond universality

can be choosen the same for all models model specific Can we calculate ?

12

Fractal dimension d

2

for power-law banded matrices

Supersymmetric virial expansion: where V. E. Kravtsov, AO, O. M. Yevtushenko, E. Cuevas , Phys. Rev. B

82

, 161102(R) (2010) V. E. Kravtsov, AO, O. M. Yevtushenko, J. Phys. A

44

, 305003 (2011) 13

Weak multifractality

How one can calculate ? Mapping onto the non-linear σ-model Perturbative expansion in the regime 14

Non-universal contributions to the fractal dimensions

Anomalous fractal dimensions Power-law ensemble Ultrametric ensemble couplings of the sigma-models ? models are different I. Rushkin, AO, Y. V. Fyodorov, J. Stat. Mech. L03001 (2011) 15

Outline

1. Power-law banded matrices and ultrametric ensemble: universality of the fractal dimensions 2. Fractal dimensions: beyond the universality 3. Criticality in 2D long-range hopping model 4. Dynamical scaling: validity of Chalker’s ansatz 16

2D power-law random hopping model

critical at

Strong criticality

CFT prediction: 17 AO, I. Rushkin, E. Cuevas, arXiv:1101.2641

2D power-law random hopping model Weak criticality

Perturbative calculations in the non-linear σ-model: Propagator Non-fractal wavefunctions: AO, I. Rushkin, E. Cuevas, arXiv:1101.2641

18

Outline

1. Power-law banded matrices and ultrametric ensemble: universality of the fractal dimensions 2. Fractal dimensions: beyond the universality 3. Criticality in 2D long-range hopping model 4. Dynamical scaling: validity of Chalker’s ansatz 19

Spectral correlations

Strong multifractality: Strong overlap of two infinitely sparse fractal wave functions! 20 J.T. Chalker and G.J.Daniell, Phys. Rev. Lett

. 61

, 593 (1988)

Return probability

Strong multifractality : V. E. Kravtsov, AO, O. M. Yevtushenko, E. Cuevas , Phys. Rev. B

82

, 161102(R) (2010) V. E. Kravtsov, AO, O. M. Yevtushenko, J. Phys. A

44

, 305003 (2011) 21

Summary

• • • • • Two critical random matrix ensembles: the power-law random matrix model and the ultrametric model Analytical results for the multifractal dimensions in the regimes of the strong and the weak multifractality Universal and non-universal contributions to the fractal dimensions Non-fractal wavefunctions in 2D critical random matrix ensemble Equivalence of the spectral and the spatial scaling exponents 22

Fractal dimensions in the ultrametric ensemble

Anomalous exponents: Symmetry relation: Y. V. Fyodorov, AO and A. Rodriguez, J. Stat. Mech., L12001 (2009) A. D. Mirlin et. al., Phys. Rev. Lett.

97

, 046803 (2007) 23

2D power-law random hopping model

AO, I. Rushkin, E. Cuevas, arXiv:1101.2641

24